# Class 9 Math Congruent Triangles 3. Fill in the blanks to make the sentences true sentences:(v) In a right-angled triangle side opposite to right angle is called

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##### 3. Fill in the blanks to make the sentences true sentences:(v) In a right-angled triangle side opposite to right angle is called

4. Find the value of unknowns for the given congruent triangles.

2. From a point on the line bisector of an angle perpendiculars are drawn to the arms of the angle. Prove that these perpendiculars are equal in measure.

1. \mathrm{ABC} is an isosceles triangle. \mathrm{D} is the mid-point of base \overline{\mathrm{BC}} . Prove that \overline{\mathrm{AD}} bisects \angle \mathrm{A} and \overrightarrow{\mathrm{AD}} \perp \overrightarrow{\mathrm{BC}} .

2. Prove that a point which is equidistant from the end points of a line segment is on the right bisector of the line segment.

3. Fill in the blanks to make the sentences true sentences:(vi) The sum of the measures of acute angle of a right triangle is

(iv) How many acute angles are there in an acute angled triangle?(a) 1(b) 2(c) 3(d) not more than 2 .

1. In the figure \overline{\mathrm{AB}} \cong \overline{\mathrm{DC}} \overline{\mathrm{AD}} \cong \overline{\mathrm{BC}} . Prove that \angle \mathrm{A} \cong \angle \mathrm{C} \angle \mathrm{ABC} \cong \angle \mathrm{ADC} .

1. In the given figure m \overline{\mathrm{AB}}=m \overline{\mathrm{CB}} and \angle \mathrm{A} \cong \angle \mathrm{C} prove that \triangle \mathrm{ABD} \cong \triangle \mathrm{CBE}

3. \mathrm{ABC} is a triangle in which m \angle \mathrm{A}=25^{\circ} m \angle \mathrm{B}=45^{\circ} and \overline{\mathrm{CD}} \perp \overline{\mathrm{AB}} . Prove that \triangle \mathrm{DBC} is an isosceles \Delta .

3. P Q R S is a square. X Y and Z are the mid-points of \overline{P Q} \overline{Q R} and \overline{\mathrm{RS}} respectively. Prove that \triangle \mathrm{PXY} \cong \Delta \mathrm{SZY} .

(ii) In \triangle \mathrm{ABC} if \angle \mathrm{A} \cong \angle \mathrm{B} then the bisector of angle divides the triangle into congruent triangles:(a) \angle \mathrm{A} (b) \angle B (c) \angle C (d) any one of its angles.

4. Prove that the median bisecting the base of an isosceles triangle bisect the vertical angle and is perpendicular to the base.

3. If \Delta \mathrm{ABC} \cong \Delta \mathrm{LMN} then find the unknown x .

2. If \triangle \mathrm{ABC} \cong \Delta \mathrm{LMN} then(iii) \mathrm{m} \angle \mathrm{A} \cong \ldots \ldots \ldots \ldots

2. Identify true and false statement in the following:(i) The sum of the measure of all angles in an rilateral is 360^{\circ} .

2. From a point on the bisector of an angle perpendiculars are drawn to the arms of the angle. Prove that these perpendiculars are equal in measure.

1. \mathrm{ABC} is a triangle in which m \angle \mathrm{A}=35^{\circ} and m \angle \mathrm{B}=100^{\circ} \overline{\mathrm{BD}} \perp \overline{\mathrm{AC}} . Prove that \triangle B D C is an isosceles triangle.

Example 2If the bisector of an angle of a triangle bisects the side opposite to it the triangle is isosceles.

2. Identify true and false statement in the following:(iv) In ân îsosceles triangle corresponding angles and correspoinding sides are equal in measure.

3. In the given figure we have \triangle A C E \cong \triangle B D E such that: m \overline{A C}=m \overline{B D}=3 \mathrm{~cm} \angle \mathrm{A}=(3 x+1)^{\circ} m \angle \mathrm{E}= (3 y-2)^{\circ} and m \angle \mathrm{B}=(x+35)^{\circ} . Find the values of x and y .

3. Fill in the blanks to make the sentences true sentences:(i) In \triangle \mathrm{ABC} \leftrightarrow \triangle \mathrm{DEF} then \overline{\mathrm{AC}} corresponds to

5. Prove that if three altitudes of a triangle are congruent then the triangle is equilatera

4. Encircle the corresponding letters a b c or d for correct answer:(i) Which of the following is not a sufficent condition for congurence of two triangles?(a) A.S.A\congA.S.A(b) H.S.H \cong H.S.H(c) S.A.A\cong S.A.A(d) A. A.A\cong A.A.A

Example 1If one angle of a right triangled triangle is of 30^{\circ} the hypotenuse is twice as long as the side opposite to the angle.

1. Prove that any two medians of an equilateral triangle are equal in measure.

2. If \triangle \mathrm{ABC} \cong \Delta \mathrm{LMN} then(i) \mathrm{m} \angle \mathrm{M} \cong

EXERCISE 10.4 1. In \triangle \mathrm{PAB} of figure \overline{\mathrm{PQ}} \perp \overline{\mathrm{AB}} and \overline{\mathrm{PA}} \cong \overline{\mathrm{PB}} prove that \overline{\mathrm{AQ}} \cong \overline{\mathrm{BQ}} and \angle \mathrm{APQ} \cong \angle \mathrm{BPQ} .

1. In the given figure \overline{\mathrm{AB}} \cong \overline{\mathrm{CB}} \angle 1 \cong \angle 2 . Prove that \triangle \mathrm{ABD} \cong \triangle \mathrm{CBE} .